Contribution of negative-energy states to the $E2\text{−}M1$ polarizability of optical clocks

We investigate the contribution of negative-energy states on the $E2$ and $M1$ polarizabilities for the ${}^{1}S_0$ and ${}^{3}P_0$ states of the Sr clock and find that they dominate the $E2\text{−}M1$ polarizability difference. Our calculated result is $-7.74\left(3.92\right)\times {10}^{-5}$ a.u., which resolves the sign inconsistency between theory and experiment. Additionally, we extend our method to other optical clocks, confirming the crucial role of negative-energy states in determining the $M1$ polarizability.

Figure 1

Comparison of the $E2\text{−}M1$ polarizability difference ${\tilde{\alpha }}^{\mathrm{QM}}/h$ (in mHz) for the Sr clock. The green data points represent experimental results, the magenta data points represent other theoretical results, and the blue data point represents our present result.

Figure 2

Sketch diagram for obtaining the positive- and negative-energy states of the Sr clock. ${\varphi }_{+}\left(\mathit{r}\right)$ and ${\varphi }_{-}\left(\mathit{r}\right)$ represent two branches of the ${\mathrm{Sr}}^{+}$ wave functions, which are used to construct the configuration state wavefunctions ${\mathrm{\Phi }}_I\left(\sigma \pi JM\right)$. While ${\mathrm{\Psi }}_{+}\left(\pi JM\right)$ and ${\mathrm{\Psi }}_{-}\left(\pi JM\right)$ represent two branches of the Sr wave functions.

Figure 3

Comparison of the $M1$ polarizability difference $\mathrm{\Delta }{\alpha }^{M1}\left(\omega \right)$ (in a.u.) for other optical clocks. The blue and magenta lines represent $\mathrm{\Delta }{\alpha }^{M1}\left(\omega \right)$ with and without the contribution of negative-energy states, respectively.